Critical exponent for the global existence of solutions to a nonlinear degenerate/singular parabolic equation

We investigate a non-homogeneous nonlinear heat equation which involves degenerate or singular coefficients belonging to the $A_2$ class of functions. We prove the existence of a Fujita exponent and describe the dichotomy existence/non-existence of global in time solutions. The $A_2$ coefficient admits either a singularity at the origin or a line of singularities. In this latter case, the problem is related to the fractional laplacian, through the Caffarelli-Silvestre extension and is a first attempt to develop a parabolic theory in this setting.